s-covering maps with complete fibers

The purpose of this note is to study s-covering maps f :X→Y with compact or more generally with complete (in the given metric on X) fibers f−1(y). The question of whether f is inductively perfect has been recently negatively solved by Debs and Saint Raymond. In the first part of this paper we show that the answer is “yes” if f−1(S) is complete in the given metric on X for every convergent including its limit sequence S⊂Y. In the second part point-harmonious maps are used in considering the properties of f. We prove that if X is a Borel set of the multiplicative class α then Y is a Borel set of the multiplicative class 3+α.